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Although the barbarian was introduced in Advanced Dungeons & Dragons (1st edition) in the Unearthed Arcana supplement, they were tribal wilderness warriors more akin to rangers, and didn't have anything resembling the rage ability. It wasn't until Dungeons & Dragons 3rd edition that barbarians appeared with an ability with the actual name "Rage." However, there were earlier versions of the Rage ability that just had different names.

In The Complete Barbarian's Handbook (AD&D 2nd edition) there is the Brute class kit which provides a barbarian with the "Wild Brawl" special ability, which is introduced with, "When fighting without weapons, the Brute can propel himself into a berserk frenzy." However, "Wild Brawl" is mechanically entirely unrelated to Rage.

In the same book, the Ravager class kit provided the barbarian with the "Become Enraged" special ability, with which the "Ravager may work himself into a fighting frenzy, increasing his effectiveness in combat." Mechanically, an enraged Ravager hits easier and does more damage, is hit easier, is harder to damage, and is harder to charm.

Also in AD&D 2nd edition there is a Warrior class kit exclusive to dwarves in The Complete Book of Dwarves called the Battlerager, which is "a fearless warrior, able to create an insane rage within himself which increases his fighting ability and distorts his physical features," and therefore a dwarven barbarian in all but name. A Battlerager in "the Killing Rage" receives bonus hit points, bonuses to attack and damage, and an AC bonus, as well as being immune to charm-like effects, but may not stop fighting until there are no more enemies standing.

Similar to the Battlerager is the is the Berserker class kit for fighters in The Complete Fighter's Handbook (AD&D 2nd edition). A Berserker takes a long time to "Go Berserk", but once Berserk gains similar benefits and also can't stop fighting until every enemy is down. Notably, a Berserker isn't allowed to know their own hit points while Berserk!

The Complete Book of Dwarves was published in 1991, four years before The Complete Barbarian's Handbook; but The Complete Fighter's Handbook was published two years earlier in 1989, making the Berserker the first barbarian-like† character with an ability that is recognisable as a "rage" ability.

†Bear in mind that before D&D3, the actual barbarian class was restricted to humans, so fighter class variants like the Berserker and Battlerager was the official way of playing a non-human character that fulfilled the same class role despite not bearing the name "barbarian."

Although the barbarian as a class was not introduced until AD&D, the idea of rage does predate that publication.

The Dragon issue #3 (October 1976) had rules for a "New D&D Subclass: The Berserker" (by Jon Pickens). This class had an ability called Berserking.

From that article:

BERSERKING

There is only a chance of a character
berserking when the referee deems the
conditions are suitable for the arousal of
battle lust. The basic chance is 10%
for a character to go berserk. A
berserk attack may only be triggered
once per melee.

Disadvantages of berserking

A berserker may not withdraw from melee while berserk.

A berserker remains berserk for as many melee rounds as he has constitution points, but the attacks at -2/-2 for the rest of the melee. If in wereshape, he will maintain it for the duration of the melee.

If still berserk at the end of the melee, he has a 10% chance of turning on his friends until killed or the fit wears off. Deduct 1% per level to a minimum of 1% (0% for Clanmasters).

The berserker must rest a complete turn after each fit.

A Barbarian class was also published in White Dwarf #4 (Dec 1977) and reprinted in The Best of White Dwarf. It had a mechanism called First-strike Ferocity:

First attack Ferocity: Barbarians of
strenqth lO+ always hit at +1 beeause
at their natural ferocity. In
addilion, if strength is 13+ and
dexterity at least average then they
have 75% chance in any melee in which
they gain lhe initiative of using
First-attack Ferocity, This is their
chance of whipping themselves up into
such a frenzy that their first attack
only will be carried out at a bonus
and do double damage or better.

All barbarians who succeed in
striking with first-attack ferocity
automatically do at least double
damage if they roll 75 or less on
percentage dice. Before the first
attack is make, roll percentile dice
and consult the following table:

[...]

Thus a barbarian capable of
first-attack always gains at least +1
tOn attack dice even if the die-roll
is too high for double damage or
greater. Remember though, this bonus
only applies to the first melee round,
and only if the barbarian has the
initiative. After this first round he
will attack normally; the rule about
barbarians of strength 10+ always
hitting at +1 is cancelled,

A barbarian is entitled to only one
round of first-attack ferocity per
battle. If he switches opponents in
battle he does not gain any
first-attack bonus on his new
opponent. Similarly, if the
barbarian's first·attack ferocity
attempt misses, he does not get
another attempt in the next round,

There was also a version of the barbarian in an article by Gygax in The Dragon #63 (July 1982, predating Unearthed Arcana by a couple years) but, like the Unearthed Arcana version, it lacked a Rage ability.